An angle is formed by joining the endpoints of two half-lines called rays. The side you measure to is called the terminal side. Angles measured counterclockwise are given a positive sign and angles measured clockwise are given a negative sign. Negative Angle This is a clockwise rotation. Positive Angle This is a counterclockwise rotation. Initial Side The side you measure from is called the initial side.
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It's Greek To Mel. It is customary to use small letters in the Greek alphabet to symbolize angle measurement. alpha 0 theta beta phi gamma delta
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We can use a coordinate system with angles by putting the initial side along the positive x-axis with the vertex at the origin. Quadrant Quadrant I angle 9 positive initial Side 0 negative If the terminal side is along an axis it is called a quadrantal angle. 1 angle Quadrant angle We say the angle lies in whatever quadrant the terminal side lies in.
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We will be using two different units of measure when talking about angles: Degrees and Radians 9 = 3600 If we start with the initial side and go all of the way around in a counterclockwise direction we have 360 degrees If we went 1/4 of the way in a clockwise direction the angle would measure -900 _ 900 900 You are probably already familiar with a right angle that measures 1/4 of the way around or 900 Let's talk about degrees first. You are probably already somewhat familiar with degrees.
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What is the measure of this angle? You could measure in the positive direction and go around another rotation which would be another 3600 O = 450 You could measure in the positive direction You could measure in the negative direction 3600 + 450 = 405 There are many ways to express the given angle. Whichever way you express it, it is still a Quadrant I angle since the terminal side is in Quadrant I.
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If the angle is not exactly to the next degree it can be expressed as a decimal (most common in math) or in degrees, minutes and seconds (common in surveying and some navigation). 1 degree = 60 minutes 1 minute = 60 seconds 048 30 d e g rees seconds minutes To convert to decimal form use conversion fraction These are fractions where then erator = denominator but two different units. Put u on top you want to convert to and put unit on ttom you want to get rid of. 30 1' Let's convert the = 0.5' seconds to 60 minutes
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1 degree = 60 minutes 1 minute = 60 seconds 9 = 25048'30" = 25048.5' = 25.8080 Now let's use another conversion fr tion to get rid of minutes. 48.5 1 0 =.8080
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Another way to measure angles is using what is called radians. Given a circle of radius r with the vertex of an angle as the center of the circle, if the arc length formed by intercepting the circle with the sides of the angle is the same length as the radius r, the angle measures one radian. arc length is also r radius of circle is r itial side This angle measures 1 radian
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Arc length s of a circle is found with the following formula: IMPORTANT: ANGLE : 4-- MEASURE MUST BE IN RADIANS TO USE FORMULA! arc length radius measure of angle Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.52 radian. arc length to find is in black 0 = 0.52 3 S = 3 0.52 = 1.56m What if we have the measure of the angle in degrees? We can't use the formula until we convert to radians, but how?
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We need a conversion from degrees to radians. We could use a conversion fraction if we knew how many degrees equaled how many radians. Let's start with the arc length formula cancel the r's 2 It radians S = 3600 If we look at one revolution around the circle, the arc length would be the circumference, Recall that circumference of a circle is This tells us that the radian measure all the way around is 21t, All the way around in degrees is 3600.
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2 radians = 3600 TC radians = 1800 Convert 300 to radians using a conversion fraction. IT radians o 180 The fraction can be reduced by 2, This would be a simpler conversion fraction, Can leave with nor use r button on radians 0.52 your calculator for decimal. 6 Convert It/3 radians to degrees using a conversion fraction. 1800 = 600 radiansâ€”
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Area Of a Sector of a Circle The formula for the area of a sector of a circle (shown in red here) is derived in your textbook. It is: Again Omust be in RADIANS so if it is in de rees you must convert to radians to use the formula. A = â€” r20 2 Find the area of the sector if the radius is 3 feet and 500 radians = 0.873 radians A = 50 180 1 2 = 3.77 sq ft
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A Sense of Angle Sizes 300 = 900 = See if you can guess the size of these angles first in degrees and then in radians. 1200 = 1500= 600 = 1800 1350= You will be working so much with these angles, you should know them in both degrees and radians.
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